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Assortment and Price Optimization

Under the Two-Stage Luce model

Alvaro Flores∗Gerardo Berbeglia†Pascal Van Hentenryck‡

Monday 22nd April, 2019

Abstract

This paper studies assortment and pricing optimization problems under the Two-Stage Luce

model (2SLM), a discrete choice model introduced by Echenique and Saito (2018) that gener-

alizes the multinomial logit model (MNL). The model employs an utility function as in the the

MNL, and a dominance relation between products. When consumers are oﬀered an assortment

S, they ﬁrst discard all dominated products in Sand then select one of the remaining products

using the standard MNL. This model may violate the regularity condition, which states that

the probability of choosing a product cannot increase if the oﬀer set is enlarged. Therefore, the

2SLM falls outside the large family of discrete choice models based on random utility which con-

tains almost all choice models studied in revenue management. We prove that the assortment

problem under the 2SLM is polynomial-time solvable. Moreover, we show that the capacitated

assortment optimization problem is NP-hard and but it admits polynomial-time algorithms for

the relevant special cases cases where (1) the dominance relation is attractiveness-correlated

and (2) its transitive reduction is a forest. The proofs exploit a strong connection between

assortments under the 2SLM and independent sets in comparability graphs. Finally, we study

the associated joint pricing and assortment problem under this model. First, we show that well

known optimal pricing policy for the MNL can be arbitrarily bad. Our main result in this sec-

tion is the development of an eﬃcient algorithm for this pricing problem. The resulting optimal

pricing strategy is simple to describe: it assigns the same price for all products, except for the

one with the highest attractiveness and as well as for the one with the lowest attractiveness.

1 Introduction

Revenue Management (RM) is the managerial practice of modifying the availability and the prices

of products in order to maximise revenue or proﬁt. The origin of this discipline dates back to

the 1970’s, following the deregulation of the US airline market. A large volume of research has

been devoted to this area over the last 45 years, with successful results in many industries ranging

∗College of Engineering & Computer Science, Australian National University, Australia.

†Melbourne Business School, The University of Melbourne, Australia.

‡H. Milton Stewart School of Industrial and Systems Engineering Georgia Institute of Technology, USA.

1

from airlines, hospitality, retailing, and others (McGill and van Ryzin, 1999; K¨ok, Fisher, and

Vaidyanathan, 2005; Vulcano, van Ryzin, and Chaar, 2010).

Two main problems lay in the core of RM theory and practice: the optimal assortment problem,

and the pricing problem. The optimal assortment problem consists of selecting a subset of products

to oﬀer customers in order to maximize revenue. Consider, for example, a retailer with limited space

allocated to mobile phones. If the store has more than 500 mobile phones that can be acquired

through its distributors (in various combinations of brands and sizes) and the mobile phone aisle

has capacity to ﬁt 50 phones on the shelves, the store manager has to decide which subset of

products to oﬀer given the product costs and the customer preferences.

In order to solve the assortment problem we need a model to predict how customers select

products when they are presented with a set of alternatives. Most models of discrete choice theory

postulate that consumers assign an utility to each alternative and given an oﬀer set, they would

choose the alternative with maximum utility. Diﬀerent assumptions on the distribution of the

utilities lead to diﬀerent discrete choice models: Celebrated examples include the multinomial logit

(MNL) (Luce, 1959), the mixed multinomial logit (MMNL) (Daly and Zachary, 1978), and the

nested multinomial logit (NMNL) (Williams, 1977).

The multinomial logit model (MNL), also known as the Luce model, is widely used in discrete

choice theory. Since the model was introduced by Luce (1959), it was applied to a wide variety of

demand estimation problems arising in transportation (McFadden, 1978; Catalano, Lo Casto, and

Migliore, 2008), marketing (Guadagni and Little, 1983; Gensch, 1985; Rusmevichientong, Shen, and

Shmoys, 2010), and revenue management (Talluri and Van Ryzin, 2004; Rusmevichientong, Shen,

and Shmoys, 2010). One of the reasons for its success stems from its small number of parameters

(one for each product): This allows for simple estimation procedures that generally avoids over

ﬁtting problems even when there is limited historical data (McFadden, 1974). However, one of the

ﬂaws of the MNL is the property known as the Independence of Irrelevant Alternatives(IIA), which

states that the ratio between the probabilities of choosing elements xand yis constant regardless

of the oﬀered subset. This property does not hold when products cannibalize each other or are

perfect substitutes (Ben-Akiva and Lerman, 1985; Debreu, 1960; Anderson, Depalma, and Thisse,

1992).

Several extensions to the MNL model have been introduced to overcome the IIA property and

some of its other weaknesses; They include the nested multinomial logit and the latent class MNL

model. These models however do not handle zero-probability choices well. Consider two products

aand b: The MNL model states that the probability of selecting aover bdepends on the relative

attractiveness of acompared to the attractiveness of b. Consider the case in which bis never

selected when ais oﬀered. Under the MNL model, this means that bmust have zero attractiveness.

But this would prevent bfrom being selected even when ais not oﬀered in an assortment.

On the other hand, the pricing problem amounts to determine the prices that a company should

oﬀer, in order to best meet its objectives (proﬁt maximization, revenue maximization, market share

maximization, etc.), while taking into consideration how customers will respond to diﬀerent prices

2

and the interaction between price and the intrinsic features that each product possess.

This paper considers both problems mentioned before, for the case when customers follow the

Two-Stage Luce model (2SLM). The 2SLM was recently introduced by Echenique and Saito (2018)

and unlike the MNL, it allows for violations to the IIA property and regularity (Berbeglia and

Joret, 2017). The Two-Stage Luce model generalizes the MNL by incorporating a dominance (anti-

symmetric and transitive) relation among the alternatives. Under such relationship, the presence

of an alternative xmay prevent another alternative yfrom being chosen despite the fact that both

are present in the oﬀered assortment. In this case, alternative xis said to dominate alternative y.

However, when xis not present, ymight be chosen with positive probability if it is not dominated

by any other product z.

An important application of the 2SLM can be found in assortment problems where there exists

a direct way to compare the products over a set of features. For illustration, consider a telecommu-

nication company oﬀering phone plans to consumers. A plan is characterized by a set of features

such as price per month, free minutes in peak hours, free minutes in weekends, free data, price for

additional data, and price per minute to foreign countries. Given two plans xand y, we say that

plan xdominates plan y, if the price per month of xis less than that of y, and xis at least as good

as yin every single feature. In the past, the company oﬀered consumers a certain set of plans St

each month tsuch that no plan in Stis dominated by another plan (in St). The oﬀered plans how-

ever were diﬀerent each month. Using historical data and assuming that consumers preferences can

be approximated using a multinomial logit, it is possible to perform a robust estimation procedure

to obtain the parameters of such MNL model. Once the parameters are obtained, the assortment

problem consists in ﬁnding the best assortment of phones plans S∗to maximize the expected rev-

enue. A natural constraint in this problem consisting in enforcing that every phone plan oﬀered

in S∗cannot be dominated by any other. Section 4 shows that the problem discussed here can

be modelled using the 2SLM and thus solving this problem is reduced to solving an assortment

problem under the 2SLM.

2 Contributions

The ﬁrst key contribution is to show that the assortment problem can be solved in polynomial time

under the 2SLM. The proof is built upon two unrelated results in optimization: the polynomial-

time solvability of the maximum-independent set in a comparability graph (M¨ohring, 1985) and

a seminal result by Megiddo (1979) that provides an algorithm to solve a class of combinatorial

optimization problems with rational objective functions in polynomial time. This is particularly

appealing since the 2SLM is one of the very few choice models that goes beyond the random utility

model and it allows violations the property known as regularity: the probability of choosing an

alternative cannot increase if the oﬀer set is enlarged. Since many decades ago, there are well-

documented lab experiments where the regularity property is violated (Huber, Payne, and Puto,

1982; Tversky and Simonson, 1993; Herne, 1997).

3

The second key contribution is to show that the capacitated assortment problem under the 2SLM

is NP-hard, which contrasts with results on the MNL. We then propose polynomial algorithms for

two interesting subcases of the capacitated assortment problem: (1) When the dominance relation

is attractiveness-correlated and (2) when the transitive reduction of the dominance relation can be

represented as a forest. The proofs use a strong connection between assortments under the 2SLM

and independent sets.

The third and ﬁnal contribution, is an in-depth study of the pricing problem under the 2SLM.

We ﬁrst note that changes in prices should be reﬂected in the dominance relation if the diﬀerences

between the resulting attractiveness are large enough. This is formalized by solving the Joint

Assortment and Pricing problem under the Threshold Luce model, where one product dominates

another if the ratio between their attractiveness is bigger than a ﬁxed threshold. Under this

setting, we show that this problem can be solved in polynomial time. The proof relies on the

following interesting facts: (1) An intrinsic utility ordered assortment is optimal; (2) the optimal

prices can be obtained in polynomial time; and (3) it assigns the same price for all products, except

for two of them, the highest and lowest attractiveness ones. Many of these results are extended to

the following cases (1) capacity constrained problems, where the number of products that can be

oﬀered is restricted and (2) position bias, where products are assigned to positions, altering their

perceived attractiveness.

The rest of the paper is organized as follows: Section 3 presents a review of the literature con-

cerning assortment optimization and pricing under variations of the Multinomial Logit. Section 4

formalizes the 2SLM and some of its properties. Section 5 proves that assortment optimization

under the 2SLM is polynomial-time solvable. Section 6 presents the results on the capacitated

version, particularly the NP-hardness of the capacitated version of the problem, but also provide

polynomial time solutions for two special cases. Section 7 present the results for pricing optimisa-

tion under the Threshold Luce model. Section 8 concludes the paper and provides future research

directions. All proofs missing from the main text, are provided in Appendix A.

3 Literature Review

Since the assortment problem and the joint assortment and pricing problem are a very active

research topic, we focus on recent results closely related with this paper and in particular, results

over the multinomial logit model (MNL) (Luce, 1959; McFadden, 1978) and its variants.

Despite the IIA property, the MNL is widely used. Indeed, for many applications, the mean

utility of a product can be modeled as a linear combination of its features. If the features capture

the mean utility associated with each product, then the error between the utilities and their means

may be considered as independent noise and the MNL emerges as a natural candidate for modeling

customer choice. In addition, the MNL parameters can be estimated from customer choice data,

even with limited data (Ford, 1957; Negahban, Oh, and Shah, 2012), because the associated estima-

tion problem has a concave log likelihood function (McFadden, 1974) and it is possible to measure

4

how good the ﬁtted MNL approximates the data (Hausman and McFadden, 1984). Moreover, it is

possible to improve model estimation when the IIA property is likely to be satisﬁed (Train, 2003).

One of the ﬁrst positive results on the assortment problem under the multinomial logit model was

obtained by Talluri and Van Ryzin (2004), where the authors showed that the optimal assortment

can be found by greedily by adding products to the oﬀered assortment in the order of decreasing

revenues, thus evaluating at most a linear number of subsets. Rusmevichientong, Shen, and Shmoys

(2010) studied the assortment problem under the MNL but with a capacity constraint limiting the

products that can be oﬀered. Under these conditions, the optimal solution is not necessarily a

revenue-ordered assortment but it can still be found in polynomial time.

Gallego, Ratliﬀ, and Shebalov (2011) proposed a more general attraction model where the

probabilities of choosing a product depend on all the products (not only the oﬀered subset as in

the MNL). This involves a shadow attraction value associated with each product that inﬂuence the

choice probabilities when the product is not oﬀered. Davis, Gallego, and Topaloglu (2013) showed

that a slight transformation of the MNL model allows for the solving of the assortment problem

when the choice probabilities follow this more sophisticated attraction model. This continues to

hold when assortments must satisfy a set of totally unimodular constraints.

The Mixed Multinomial Logit (Daly and Zachary, 1978) is an extension of the MNL model,

where diﬀerent sets of customers follow diﬀerent MNL models. Under this setting, the problem

becomes NP-hard (Bront, M´endez-D´ıaz, and Vulcano, 2009) and it remains NP-hard even for two

customer types (Rusmevichientong et al., 2014). A branch-and-cut algorithm was proposed by

M´endez-D´ıaz et al. (2014). Feldman and Topaloglu (2015) proposed methods to obtain good upper

bounds on the optimal revenue. Rusmevichientong and Topaloglu (2012) considered a model where

customers follow a MNL model and the parameters belong to a compact uncertainty set. The ﬁrm

wants to hedge against the worst-case scenario and the problem amounts to ﬁnding an optimal

assortment under this uncertainty conditions. Surprisingly, when there is no capacity constraint,

the revenue-ordered strategy is optimal in this setting. Jagabathula (2014) proposed a local-search

heuristic for the assortment problem under an arbitrary discrete choice model. Davis, Gallego,

and Topaloglu (2013) and Abeliuk et al. (2016) proposed polynomial time algorithms to solve

the assortment problem under the MNL model with capacity constraint and position bias, where

position bias means that customer choices are aﬀected by the positioning of the products in the

assortment. Recently, Jagabathula and Vulcano (2015) proposed a partial-order model to estimate

individual preferences, where preference over products are modeled using forests. They cluster the

customers in classes, each class being represented with a forest. When facing an assortment S,

customers select, following an MNL model, products that are roots of the forest projected on S.

This approach outperformed state-of-the-art methods when measuring the accuracy of individual

predictions.

Attention has also been devoted to discrete choice models to represent customer choices in

more realistic ways, including models that violate the IIA property (Ben-Akiva and Lerman, 1985).

This property does not always hold in practice (Rieskamp, Busemeyer, and Mellers, 2006), includ-

5

ing when products cannibalize each other (Ben-Akiva and Lerman, 1985). Echenique, Saito, and

Tserenjigmid (2018) identify these violations as perception priorities, and adjust probabilities to

take their eﬀects into account. Gul, Natenzon, and Pesendorfer (2014) provide an axiomatic gen-

eralization of MNL model to address the case where the products share features. Fudenberg and

Strzalecki (2015) propose an axiomatic generalization of a discounted logit model incorporating a

parameter to model the inﬂuence of the assortment size.

Customers tend to use rules to simplify decisions, and before making a purchase decision, they

often narrow down the set of alternatives to chose from, using diﬀerent heuristics to make the

decision process simpler. Several models of consider-then-choose models have been proposed in

the literature, related with attention ﬁlters, search costs, feature ﬁlters, among others, another

reasonable way to discard options, is when the diﬀerence between attractiveness is so evident, that

the less attractive alternative, even when it is oﬀered, is never picked (as in the Threshold Luce

model, Echenique and Saito (2018)). Any of the heuristics mentioned before allows the consumer

to restrict her attention to a smaller set usually referred in the literature as consideration set. This

eﬀect also provokes that oﬀered product might result having zero-probability choices.

Several models have been proposed to address the issue of zero-probability choices. Masatlioglu,

Nakajima, and Ozbay (2012) propose a theoretical foundation for maximizing a single preference

under limited attention, i.e., when customers select among the alternatives that they pay attention

to. Manzini and Mariotti (2014) incorporate the role of attention into stochastic choice, proposing a

model in which customers consider each oﬀered alternative with a probability and choose the alter-

native maximizing a preference relation within the considered alternatives. This was axiomatized

and generalized in Brady and Rehbeck (2016), by introducing the concept of random conditional

choice set rule, which captures correlations in the availability of alternatives. This concept also

provided a natural way to model substitutability and complementarity.

Payne (1976) showed that a considerable portion of the subjects in his experimental setting

use a decision process involving a consideration set. Numerous studies in marketing also validated

a consider-then-choose decision process. In his seminal work Hauser (1978) observed that most

of the heterogeneity in consumer choice can be explained by consideration sets. He shows that

nearly 80% of the heterogeneity in choice is captured by a richer model based in the combination of

consideration sets and logit-based rankings. The rationale behind this observation is that ﬁrst stage

ﬁlters eliminate a large fraction of alternatives, thus the resulting consideration sets are composed

of a few products in most of the studied categories (Belonax Jr and Mittelstaedt, 1978; Hauser and

Wernerfelt, 1990). Pras and Summers (1975) and Gilbride and Allenby (2004) empirically showed

that consumers form their consideration sets by a conjunction of elimination rules. Furthermore,

there are empirical results showing that a Two-Stage model including consideration sets better

ﬁts consumer search patterns than sequential models (De los Santos, Horta¸csu, and Wildenbeest,

2012).

Form a customer standpoint, the use for consider-then-choose models alleviate the cognitive

burden of deciding when facing too many alternatives Tversky (1972a,b); Tversky and Kahneman

6

(1974); Payne, Bettman, and Luce (1996). When dealing with a decision under limited time and

knowledge, customers often recur to screening heuristics as show in Gigerenzer and Goldstein (1996).

Psychologically speaking, customers as decision makers need to carefully balance search eﬀorts and

opportunity costs with potential gains, and consideration sets help to achieve that goal (Roberts and

Lattin, 1991; Hauser and Wernerfelt, 1990; Payne, Bettman, and Luce, 1996). Recently Jagabathula

and Rusmevichientong (2017) proposed a Two-Stage model where customers consider only the

products are contained within certain range of their willingness to pay. Aouad, Farias, and Levi

(2015) explored consider-then-choose models where each costumer has a consideration set, and a

ranking of the products within it. The customer then selects the higher ranked product oﬀered.

The authors studied the assortment problem under several consideration sets and ranking structure,

and provide a dynamic programming approach capable of returning the optimal assortment in

polynomial time for families of consideration set functions originated by screening rules Hauser,

Ding, and Gaskin (2009). Dai et al. (2014) considered a revenue management model where an

upcoming customer might discard one oﬀered itinerary alternative due to individual restrictions,

such as time of departure. Wang and Sahin (2018) studied a choice model that incorporates product

search costs, so the set that a customer considers might diﬀer from what is being oﬀered.

Multi-product price optimisation under the MNL and the NL has been studied since the models

were introduced in the literature. One of the ﬁrst results on the structure of the problem is due

to Hanson and Martin (1996), where they show that the proﬁt function for a company selling

substitutable products when customers follow the MNL model is not jointly concave in price. To

overcome this issue, in Song and Xue (2007) and later in Dong, Kouvelis, and Tian (2009), the

authors show that even when the proﬁt function is not concave in prices, it is concave in the market

share and there is a one-to-one correspondence between price and market share. Multiple studies

shown that under the MNL where all products share the same price sensitivity parameter, the

mark-up which is simply the diﬀerence between price and cost, remains constant for all products

at optimality (Anderson, Depalma, and Thisse, 1992; Hopp and Xu, 2005; Gallego and Stefanescu,

2009; Besbes and Saur´e, 2016). Furthermore, the proﬁt function is also uni-modal on this constant

quantity and it has a unique optimal solution, which can be determined by studying the ﬁrst order

conditions.

Li and Huh (2011) showed the same result for the NL model. Up to that point, all previous

results assumed an identical price sensitivity parameter for all products. Under the MNL, there

is empirical evidence that shows the importance of allowing diﬀerent price sensitivity parameters

for each product (Berry, Levinsohn, and Pakes, 1995; Erdem, Swait, and Louviere, 2002). There

is is also evidence in B¨orsch-Supan (1990) that restricting the nest speciﬁc parameters to the unit

interval results in rejection of the NL model when ﬁtting the data, thus recommending to relax

this assumption. The problem when relaxing this condition, is that the proﬁt function is no longer

concave on the market share, which complicates the optimization task. In Gallego and Wang

(2014) the authors considered a NL model with diﬀerentiated price sensitivities, and found that

the adjusted mark-up, deﬁned as price minus cost minus the reciprocal of the price sensitivity is

7

constant for all products within a nest at optimality. Furthermore, each nest also has an adjusted

next-level markup which is also invariant across nests, which reduces the original problem to a

one variable optimization problem. Additional theoretical development can be found in Rayﬁeld,

Rusmevichientong, and Topaloglu (2015); Kouvelis, Xiao, and Yang (2015) but there are restricted

to the Two-Stage nested logit model. In Huh and Li (2015) some of the results were extended to a

multi-stage nested logit model for speciﬁc settings, but also show that the equal mark-up property

fails to hold in general for products that do not share the same immediate parent node in the

nested choice structure, even when considering identical price sensitivity parameters. Li and Huh

(2011) and Gallego and Wang (2014) extend to the multi-stage NL model and show that an optimal

pricing solution can still be found by means of maximizing a scalar function.

There are some interesting results for other models that share similarities with the MNL, and

therefore are closely related with the model that we are studying. In Wang and Sahin (2018), the

authors incorporate search cost into consumer choice model. The results on this paper for the Joint

Assortment and Pricing are similar to the ones that we study in Section 7, in that many structural

results that holds at optimality for their model, are also satisﬁed in our studied case. They show

that the quasi-same price policy (that charges the same price for all products but one, the least

attractive one) was optimal for this model. Interestingly, the Joint Assortment and Pricing results

under the Threshold Luce Model has a slightly diﬀerent result: The optimal pricing is a ﬁxed price

for all products, except for the most attractive and least attractive ones. This led to a situation

where there are many possible prices, not just two.

Recently Alptekino˘glu and Semple (2016) hast studied in depth a model which was originally

due to Daganzo (1979) that assumes a negatively skewed distribution of consumer utilities. The

resulting choice probabilities have an interesting consequence in the optimal pricing policy: They

allow for variable mark-ups in optimal prices that increase with expected utilities.

The model considered in this paper is a variant of the MNL, proposed by Echenique and Saito

(2018) and called the Two-Stage Luce model ; It handles zero-probability choice by introducing the

concept of dominance, meaning that if a product xdominates a product y, then yis never selected

in presence of y. And therefore the consideration set is formed by considering only non-dominated

products in the oﬀered assortment, allowing ﬂexibility on the consideration set formation due to

the nature of the dominance relation. Once the consideration set is formed, the customer choose

according to an MNL on the remaining alternatives. In the following section we describe this model

in detail, and show some examples that highlight many practical applications for it.

4 The Two-Stage Luce model

The 2SLM (Echenique and Saito, 2018) overcomes a key limitation of the MNL: The fact that

a product must have zero attractiveness if it has zero probability to be chosen in a particular

assortment. This limitation means that the product cannot be chosen with positive probability

in any other assortment. The 2SLM eliminates this pathological situation through the concept of

8

consideration function which, given a set of products S, returns a subset of Swhere each product

has a positive probability of being selected. Let Xdenotes the set of all products and let a(x)>0

be the attractiveness of product x∈X. For notational convenience, we use axto denote the

attractiveness of product x, i.e., ax=a(x). We extend the attractiveness function to consider the

outside option, with index 0 and a0=a(0) ≥0, to model the fact that customers may not select

any product. As a result, the attractiveness function has signature a:X∪ {0} → R+. Given an

assortment A⊆X, a stochastic choice function ρreturns a probability distribution over A, i.e.,

ρ(x, A) is the probability of picking xin the assortment A. The 2SLM is a sub case of the general

Luce model presented in Echenique and Saito (2018), and independently discovered in Ahumada

and ¨

Ulk¨u (2018), which is deﬁned below.

Deﬁnition 1 (General Luce Function ∗, Echenique and Saito (2018)).A stochastic choice function

ρis called a general Luce function if there exists an attractiveness function a∪ {0}:X→R+and

a function c: 2X\ ∅ → 2X\ ∅ with c(A)⊆Afor all A⊆Xsuch that

ρ(x, A) =

ax

Py∈c(A)ay+a0if x∈c(A),

0 if x /∈A.

(1)

for all A⊆X. We call the pair (a, c) a general Luce model.

The function c(which is arbitrary) provides a way to capture the support of the stochastic choice

function ρ. As observed in Echenique and Saito (2018), there are two interesting cases worthy of

being mentioned:

1. If c(S) is a singleton for all S⊆X, then ρ(x, S) is a deterministic choice.

2. If c(S) = Sfor all S⊆X, then the 2SLM coincides with the MNL.

Two special cases of this model were provided in Echenique and Saito (2018). The ﬁrst is

the two-stage Luce model. This model restricts c, such that the c(A) represents the set of all

undominated alternatives in A.

Deﬁnition 2 (two-Stage Luce model (2SLM), Echenique and Saito (2018)).A general Luce model

(a, c) is called a 2SLM if there exists a strict partial order (i.e. transitive, antisymmetric and

irreﬂexive binary relation) such that:

c(A) = {x∈A| 6 ∃y∈A:yx}.(2)

We call dominance relation.

As a result, any 2SLM can be described by an irreﬂexive, transitive, and antisymmetric relation

that fully captures the relation between products. The second model presented in Echenique and

Saito (2018), which is a particular case of the 2SLM, is the Threshold Luce Model (TLM), where

∗The deﬁnition is slightly diﬀerent: It makes the outside option eﬀect a0explicit in the denominator.

9

they explain dominance in terms of how big the attractiveness are when compared with each other,

so cis strongly tied to a. More speciﬁcally, for a given threshold t > 0, the consideration set c(S)

for a set S⊆Xis deﬁned as:

c(S) = {y∈S| 6 ∃x∈S:ax>(1 + t)ay}.(3)

In other words, xyif and only if ax

ay>(1 + t). Intuitively, an attractiveness ratio of more than

(1 + t) means that the less-preferred alternative is dominated by the more-preferred alternative.

Observe that the relation is clearly irreﬂexive, transitive, and antisymmetric.

The dominance relation can thus be represented as a Directed Acyclic Graph (DAG), where

nodes represent the products and there is a directed edge (x, y) if and only if xy. Sets satisfying

c(S) = Sare anti-chains in the DAG, meaning that there are no arcs connecting them. For

instance, consider the Threshold Luce model deﬁned over X={1,2,3,4,5}with attractiveness

values a1= 12, a2= 8, a3= 6, a4= 3 and a5= 2, and threshold t= 0.4. We have that ijiﬀ

ai>1.4aj.

The DAG representing this dominance relation is depicted in Figure 1.

1

a1= 12

2

a2= 8

3

a3= 6

4

a4= 3

5

a5= 2

Figure 1: Example of a DAG for a General Threshold Luce model

In the following example, we show that the 2SLM admits regularity violations, meaning that it

is possible that the probability of choosing a product can increase when we enlarge the oﬀered set.

Since regularity is satisﬁed by any choice model based on random utility (RUM), this shows that

the 2SLM is not contained in the RUM class †.

Example 1. Consider the following instance of the Threshold Luce model (which is a special case

of the 2SLM). Let X={1,2,3,4}with attractiveness a1= 5, a2= 4, a3= 3 and a4= 3. Consider

t= 0.4 and the attractiveness of the outside option a0= 1. For the oﬀer set {2,3,4}, the probability

of selecting product 2 is 4/11 since no product dominates each other. However, if we add product 1

to the oﬀer set, i.e. if we oﬀer all four products, then the probability of selecting product 2 increases

to 4/10, because products 3 and 4 are now dominated by product 1.

The Two-Stage Luce Model allows to accommodate diﬀerent decision heuristics and market

scenarios by specifying the dominance relation responding to a speciﬁc set of rules. Two cases

where this can be observed are provided below.

†Observe that this implies that the 2SLM is not contained by the Markov chain model proposed by (Blanchet,

Gallego, and Goyal, 2016) since this last one belongs to the RUM class (Berbeglia, 2016).

10

Feature Diﬀerence Threshold: Assume that each product has a set of features F={1, . . . , m}.

A product xcan then be represented by a m-dimensional vector x∈Rm. Assume that the

perceived relevance of each feature kis measured by a weight νk, so that the utility perceived by

the customers can be expressed as a weighted combination of their features u(x) = Pm

k=1 νk·xk.

The dominance relation can be deﬁned as xy⇐⇒ u(x)−u(y) = Pm

k=1 νk(xk−yk)≥T,

where T > 0 is a tolerance parameter that represents how much diﬀerence a customer allows before

considering that an alternative dominates another. The dominance relation is irreﬂexive, transitive,

and antisymmetric and hence it can be used to deﬁne an instance of the 2SLM. One can easily

show that this model is a special case of the TLM.

Price levels: Suppose we have Nproducts, each product ihas kiprice levels. Let xil be product

iwith price pil attached and it corresponding attractiveness ail, we assume that for each product

iprices pik satisfy pi1< pi2< . . . , piki. Naturally, xi1xi2. . . xiki, because for the same

product the customer is going to select the one with the lowest price available. Each price level

for each product can still dominate or be dominated by other products as well, as long as the

dominance relation is irreﬂexive, transitive and antisymmetric. This setting can be modelled by

the Two-Stage Luce model in a natural way.

5 Assortment Problems Under the Two-Stage Luce model

This section studies the assortment problem for the 2SLM using the deﬁnitions and notations

presented earlier. Let r:X∪ {0} → R+be a revenue function associated with each product and

satisfying r(0) = 0. The expected revenue of a set S⊆Xis given by

R(S) = X

i∈c(S)

ρ(i, S)r(i).(4)

The assortment problem amounts to ﬁnding a set

S∗∈argmax

S⊆X

R(S)

yielding an optimal revenue of

R∗= max

S⊆XR(S).

Observe that every subset S⊆Xcan be uniquely represented by a binary vector x∈ {0,1}nsuch

that i∈Sif and only if xi= 1. Using this bijection, the search space for S∗can be restricted to

D={x∈ {0,1}n| ∀st:xs+xt≤1}

where Drepresents all the subsets satisfying S=c(S), which means that no product on Sdominates

another product in S. There is always an optimal solution S∗that belongs to Dbecause R(S) =

R(c(S)) and c(S)∈Dfor all sets Sin X. As a result, the Assortment Problem under the 2SLM

11

(AP-2SLM) can be formulated as

maximize

xPn

i=1 riaixi

Pn

i=1 aixi+a0

subject to x∈ D

(AP-2SLM)

where riand airepresent r(i) and a(i) for simplicity.

An eﬀective strategy for solving many assortment problems consists in considering revenue-

ordered assortments, which are obtained by choosing a threshold ρand selecting all the products

with revenue at least ρ. This strategy leads to an optimal algorithm for the assortment problem

under the MNL. Unfortunately, it fails under the 2SLM because adding a highly attractive product

may remove many dominated products whose revenues and utilities would lead to a higher revenue.

Example 2 (Sub-Optimality of Revenue-Ordered Assortments).Consider a Threshold Luce model

with X={1,2,3}, revenues r1= 88, r2= 47, r3= 46, attractiveness a0= 55, a1= 13, a2= 26, a3=

15 and t= 0.6. Then xyiﬀ ax>1.6aywhich gives 2 1 and 2 3. Consider the sets S⊆X

satisfying S=c(S):

S R(S)

{1}16.824

{2}15.086

{3}9.857

{1,3}22.096

The optimal revenue is given by assortment {1,3}, while the best revenue-ordered assortment

under the 2SLM is S={1}, yielding almost 24% less revenue.

To solve problem AP-2SLM, consider ﬁrst the MaxAtt problem deﬁned over the same set of

constraints. Given weights ci∈R(1 ≤i≤n), the MaxAtt problem is deﬁned as follows:

maximize

x

n

X

i=1

cixi

subject to x∈ D

(MaxAtt)

We now show that (MaxAtt) can be reduced to the maximum weighted independent set problem

in a directed acyclic graph with positive vertex weights. An independent set is a set of vertices I

such that there is no edge connecting any two vertices in I. The maximum weighted independent

set problem (MWIS) can be stated as follows:

Deﬁnition 3. Maximum Weighted Independent Set Problem: Given a graph G= (V, E ) with a

weight function w:V→R, ﬁnd an independent set I∗∈argmaxI∈I Pi∈Iw(i), where Iis the set

of all independent sets.

12

Recall that the dominance relation can be represented as a DAG Gwhich includes an arc (u, v )

whenever uv. As a result, the condition x∈ D implies that any feasible solution to (MaxAtt)

represents an independent set in Gand maximizing Pn

i=1 cixiamounts to ﬁnding the independent

set maximizing the sum of the weights. Since the dominance relation is a partial order, the DAG

representing the dominance relation is a comparability graph. The following result is particularly

useful.

Theorem 1 (M¨ohring (1985)).The maximum weighted independent set is polynomially-solvable

for comparability graphs with positive weights.

We are ready to present our ﬁrst result.

Lemma 1. (MaxAtt)is polynomial-time solvable.

Proof. We ﬁrst show that we can ignore those products with a negative weight. Let ˆ

X={i∈X|

ci>0}and ˆ

D={x∈ {0,1}n| ∀s, t ∈ˆ

X, s t:xs+xt≤1}. Solving (MaxAtt) is equivalent to

solving:

maximize

xX

i∈ˆ

X

cixi

subject to x∈ˆ

D

(Reduced MaxAtt)

Indeed, consider an optimal solution x∗to Problem MaxAtt and assume that there exists i∈X

such that ci<0 and x∗

i= 1. Deﬁne ˆxlike x∗but with ˆxi= 0. ˆxhas a strictly greater value for

the objective function in Reduced MaxAtt than x∗has, and is feasible since setting a component

to zero cannot violate any constraint (i.e., ˆx∈ D). This contradicts the optimality of x∗. Now

Problem Reduced MaxAtt can be reduced to solving an instance of Problem MWIS in a DAG with

positive weights that corresponds to the dominance relation. This DAG is a comparability graph

and the result follows from Theorem 1.

The next step in solving the assortment problem under the 2SLM relies on a result by Megiddo

Megiddo (1979). Let Dbe a domain deﬁned by some set of constraints and consider Problem A

maximize

x

n

X

i=1

cixi

subject to x∈D

(A)

and its associated Problem B:

maximize

x

a0+Pn

i=1 aixi

b0+Pn

i=1 bixi

subject to x∈D.

(B)

Using this notation, Megiddo’s theorem can be stated as follows.

13

Theorem 2 (Megiddo (1979)).If Problem Ais solvable within O(p(n)) comparisons and O(q(n))

additions, then Problem Bis solvable in O(p(n)(q(n) + p(n))) time.

We are now in position to state our main theorem of this section.

Theorem 3. The assortment problem under the Two-Stage Luce model is polynomial-time solvable.

Proof. Recall that the assortment problem under the 2SLM (AP-2SLM) can be formulated as

maximize

xPn

i=1 riaixi

Pn

i=1 aixi+a0

subject to x∈ D

(5)

where D={x∈ {0,1}n| ∀st:xs+xt≤1}.

The problem of maximizing the numerator in (5) is exactly the MaxAtt problem. By Lemma

1, this is polynomial-time solvable. Now observe that (5) (i.e., problem AP-2SLM) can be seen as

a Problem B. Therefore, by Theorem 2, the assortment problem under the 2SLM is solvable in

polynomial time.

In addition to solving the assortment problem under the 2SLM, Theorem 3 is interesting in that it

solves the assortment problem under a Multinomial Logit with a speciﬁc class of constraints. It can

be contrasted with the results by Davis, Gallego, and Topaloglu (2013), where feasible assortments

satisfy a set of totally unimodular constraints. They show that the resulting problem can be solved

as a linear program. However, the 2SLM introduces constraints that are not necessarily totally

unimodular as we now show.

Example 3. Consider X={1,2,3,4}and 1 3,14,23,24,and 3 4. The constraint

matrix that deﬁnes the feasible space (D) for this instance is:

M=

1 0 1 0

1 0 0 1

0 1 1 0

0 1 0 1

0 0 1 1

where each row represents a constraint xu+xv≤1. meaning that just one end of the edge can be

selected at the time. Camion (1965) proved that Mis totally unimodular if and only if, for every

(square) Eulerian submatrix A of M,Pi,j aij ≡0 (mod 4). Consider the sub-matrix corresponding

to the ﬁrst, second, and ﬁfth rows and the ﬁrst, third, and fourth columns

N=

110

101

011

14

Matrix Nis eulerian (The sums of every element on each row or on each column is a multiple of

2). But the sum of all elements of Nis 6 6≡ 0 (mod 4) and hence Mis not totally unimodular.

We close this section by explaining how our results can be extended to a more general setting.

Gallego, Ratliﬀ, and Shebalov (2014) proposed the general attraction model (GAM) to describe

customer behaviour, that alleviates some deﬁciencies of the MNL. More speciﬁcally, the intuition

behind this choice model is that whenever a product is not oﬀered, then its absence can potentially

increase the probability of the no-purchase alternative, as consumers can potentially look for the

product elsewhere, or at a later time. To achieve this eﬀect, for each product jthe model considers

two diﬀerent weights: vjand wj, usually with 0 ≤wj≤vj. If product jis oﬀered, then its

preference weight is vj. But if jis not oﬀered, then the preference weight of the outside option is

increased by wj. For all j∈X, let evj=vj−wjand ˜v0=v0+Pk∈Xwk. Using this notation, the

probabilities associated with the GAM model can be recovered by means of the following equation:

ρ(j, S) =

vj

Pi∈Sevj+ev0if j∈S,

0 if j /∈S.

(6)

Observe that the resulting assortment problem will has the same functional form than problem

AP-2SLM, with a slight modiﬁcation on the coeﬃcients in the denominator. Thus, we can apply the

same solution technique described in Theorem 3 to ﬁnd the optimal assortment for the GAM.

6 The Capacitated Assortment Problem

In many applications, the number of products in an assortment is limited, giving rise to capacitated

assortment problems. Let C(1 ≤C≤n) be the maximum number of products allowed in an

assortment. The Capacitated Assortment Problem under the Two-Stage Luce Model (C2SLMAP) is

given by

maximize

xPn

i=1 riaixi

Pn

i=1 aixi+a0

subject to x∈ DC

(C2SLMAP)

where DC={x∈ {0,1}n| ∀(s, t)∈ R xs+xt≤1∧Pn

i=1 xi≤C}. As before, it is useful to deﬁne

its capacitated maximum-attractiveness counterpart (C-MaxAtt), i.e.,

maximize

x

n

X

i=1

cixi

subject to x∈ DC

(C-MaxAtt)

This section ﬁrst proves that the capacitated assortment problem under the 2SLM is NP-hard. The

reduction uses the Maximum Weighted Budgeted Independent Set (MWBIS) problem proposed by

Bandyapadhyay (2014) which amounts to ﬁnding a maximum weighted independent set of size not

greater than C. Kalra et al. (2017) showed that Problem (MWBIS) is NP-hard for bipartite graphs.

15

Theorem 4. Problem (C2SLMAP)is NP-hard (under Turing reductions).

It is interesting to mention that Problem (C-MaxAtt) is equivalent to ﬁnding an anti-chain of

maximum weight among those of cardinality at most C. This problem (MWLA) was proposed by

Shum and Trotter (1996) and its complexity was left open, but the above results show that it is

also NP-hard. Bandyapadhyay (2014) studied Problem (MWBIS) for various types of graphs (e.g.,

trees and forests), but the dominance relation of the 2SLM can never be a tree since it is transitive

(unless we consider a graph with a single vertex).

In light of this NP-hardness result, the rest of this section presents polynomial-time algorithms

for two special cases of the dominance relation.

6.1 The Two-Stage Luce model over Tree-Induced Dominance Relations

Let Rbe the transitive reduction of the irreﬂexive, antisymmetric, and transitive relation . This

section considers the capacitated assortment problem when the relation Rcan be represented as

a tree. Without loss of generality, we can assume that the tree contains all products. Otherwise, we

can add another product with zero weight that dominates all original products. This new product

will be the root of the tree and the products not in the original tree will be the children of the root.

Similarly, the same transformation applies to the case when Ris a forest. Here all the trees in

the forest will be children of the new product.

We show how to solve Problem (C-MaxAtt). The result follows again by applying Megiddo’s

theorem. The ﬁrst step of the algorithm simply removes all products with negative weight: Their

children can be added to the parent of the deleted vertex. The main step then solves (C-MaxAtt)

bottom-up using dynamic programming from the leaves. For simplicity, we present the recurrence

relations to compute the weight of the optimal assortment. It is easy to recover the optimal

assortment itself. The recurrence relations compute two functions:

1. A(k, c) which returns the weight of an optimal assortment using product kand its descendants

in the tree representation of Rfor a capacity c;

2. A+(S, c) which, given a set Sof vertices that are children of a vertex k, returns the weight

of an optimal assortment using the products in Sand their descendants for a capacity c.

The key intuition behind the recurrence is as follows. If vis a vertex and v1and v2are two of

its children, v1does not dominate v2or any of its descendants. Hence, it suﬃces to compute the

best assortments producing A(v1,0),...,A(v1, C ) and A(v2,0),...,A(v2, C) and to combine them

optimally. The recurrence relations are deﬁned as follows (v∈Xand 1 ≤c≤C):

A(v, 0) = 0;

A(v, c) = max(cv,A+(children(v), c));

16

and

A+(∅,0) = 0;

A+(S, c) = max

n1,n2≥0

n1+n2=c

A+(S\ {e}, n1) + A(e, n2) where e= argmax

i∈S

ci.

where children(p) denotes the children of product pin the tree. Note that A+(S, c) is computed

recursively to obtain the best assortment from the products in Sand their descendants. Using

these recurrence relation, the following Theorem follows:

Theorem 5. Let a dominance relation whose relation Ris a tree containing all products. The

capacitated assortment problem under the 2SLM and is polynomial-time solvable.

6.2 The Attractiveness-Correlated Two-Stage Luce model

The second special case considers a dominance relation that is correlated with attractiveness.

Deﬁnition 4 (Attractiveness-Correlated Two-Stage Luce model).A Two-Stage Luce model is

attractiveness-correlated if the dominance relation satisﬁes the following two conditions:

1. If xy, then ax> ay.

2. If xyand az> ax, then zy.

The ﬁrst condition simply expresses that product xcan only dominate product yif the attractive-

ness of xis greater than the attractiveness of y. The second condition ensures that, if xdominates

y, then any product whose attractiveness is greater than xalso dominates y. The induced domi-

nance relation is irreﬂexive, anti-symmetric, and transitive. A particular case of this model, is the

Threshold Luce model.

When customers follow the Threshold Luce model, they form their consideration sets based on

the attractiveness of products. Without loss of generality, we can assume a1≥a2≥. . . ≥an,

unless stated otherwise. For a set S, the associated consideration set c(S) may be a proper subset

of S, but for the purpose of assortment optimization, we don’t have incentives to oﬀer sets including

products that are not even consider by customers, so we can restrict our search for optimal solutions

to sets where c(S) = S. A necessary and suﬃcient condition for this to happen is maxi∈Sai

mini∈Sai≤1 + t.

Meaning that largest ratio between attractiveness is not greater than 1+t, so no dominance relation

appears.

The ﬁrm now needs to carefully balance the inclusion of high-attractiveness products and their

prices to maximize the revenue. In the following example we show that revenue ordered assortments

are not optimal under the Threshold Luce Model. In fact, this strategy can be arbitrarily bad.

Example 4 (Revenue ordered assortments are not optimal).Consider the following product con-

ﬁguration. Let N+ 1 products, with prices p1for the ﬁrst product, and αp1for the rest of them,

with α < 1. The attractiveness for all products is a1for the ﬁrst product and γa1for all the rest,

17

such as in the presence of product 1, all the rest of the products are ignored. To complete the

set up, let a0the attractiveness of the outside option. The best revenue ordered assortment is to

consider product 1, given a revenue of:

R0=R({1}) = p1a1

a1+a0

But, if Nis big enough (at least bigger than 1

αγ ), is more proﬁtable to show SN=X\ {1},

resulting in a revenue of:

R∗=R(SN) = N·αp1γa1

N·αγa1+a0

Now, if we calculate the ratio if this two values, R0and R∗and let Ntend to inﬁnity we have:

R0

R∗= lim

N→∞

p1a1

a1+a0

N·αp1γa1

N·αγa1+a0

R0

R∗= lim

N→∞

p1a1

a1+a0

·N·αγa1+a0

N·αp1γa1

R0

R∗=a1

a1+a0

(7)

Observe that this last expression is the market share of oﬀering just product 1, which can

be arbitrarily bad by either making a1as small as desired, or making the outside option more

attractive.

The capacitated assortment optimization can be solved in polynomial time under the Attractiveness-

Correlated Two-Stage Luce model. Consider an assortment whose product with the largest attrac-

tiveness is k. This assortment cannot contain any product dominated by k. Moreover, if k1and k2

are two other products in this assortment, then k1cannot dominate k2since kwould also dominate

k2. As a result, consider the set

Xk={i∈X|ai≤ak&k6 i}.

No product in Xkdominates any other product in Xkand hence the C2SLMAP reduces to a tradi-

tional assortment problem under the MNL. This idea is formalized in Algorithm 1, where CMLMAP

is a traditional algorithm for the MNL. The algorithm considers each product in turn and the

products that it does not dominate and applies a traditional capacitated assortment optimization

under the MNL. The best such assortment is the solution to the capacitated assortment under the

attractiveness-correlated 2SLM.

18

Algorithm 1: Capacitated Assortment Optimization under the Attractiveness-Correlated

2SLM.

Data: X, , r, a

Result: Optimal Assortment S∗

R(S∗)=0for k= 1, . . . , n do

Xk={i∈X|ai≤ak&k6 i}

Sk=CMLMAP(Xk, r, a)

if R(Sk)> R(S∗)then

S∗=Sk

end

end

return S∗

Theorem 6. C2SLMAP can be solved in polynomial time for Attractiveness-Correlated instances.

Proof. To show correctness, it suﬃces to show that the optimal assortment must be a subset of

one of the Xk(1 ≤k≤n). Let Abe the optimal assortment and assume that kis its product

with the largest attractiveness (break ties randomly). Amust be included in Xksince otherwise

it would contain a product xsuch that kx(contradicting feasibility) or such that a(x)>

a(k) (contradicting our hypothesis). The correctness then follows since there is no dominance

relationship between any two elements in each of Xk. The claim of polynomial-time solvability

follows from the availability of polynomial-time algorithms for the assortment problem under the

MNL and the fact that are exactly ncalls to such an algorithm.

7 Joint Assortment and Pricing under the Threshold Luce model

The previous sections provides solutions to the Assortment Optimization problem under the Two-

Stage Luce model. This section aims at determining how to assign prices to products in order to

maximise the expected revenue. It studies the Joint Assortment and Pricing Problem under the

Threshold Luce model, by making the attractiveness of each product dependent upon its price.

Let p= (p1, . . . , pn) be the price vector, where such that pi∈R+∪ {∞} represents the price of

product i. Since the price will aﬀect the attractiveness aiof product i, the presentation makes this

dependency explicit by writing ai(pi) whose form in this paper is speciﬁed by

ai(pi) = exp(ui−pi) (8)

where uiis the intrinsic utility of product iand the value vi=ui−piis called the net utility of

product i. Assigning an inﬁnite price to a product is equivalent to not oﬀering the product, as the

attractiveness, and therefore the probability of selecting the product, becomes 0. Without loss of

generality, products are indexed in a decreasing order by intrinsic utility.

19

i uipiai(pi)p0

iai(p0

i)

1 ln(10) ln(3) 3.3 ln(4) 2.5

2 ln(8) ln(3) 2.6 ln(4) 2

3 ln(6) ln(3) 2 ln(3) 2

4 ln(3) ln(3) 1 ln(2) 1.5

Table 1: Summary of utilities, prices and attractiveness for the two proposed scenarios.

1

a1(p1)=3.3

2

a2(p2)=2.6

3

a3(p3)=2

4

a4(p4) = 1

Figure 2: The DAG for the ﬁrst scenario where all prices are ﬁxed to ln(3) and the threshold is

t= 0.5. Product 1 dominates products 3 and 4, and product 2 dominates product 4.

1

a1(p0

1)=2.5

2

a2(p0

2) = 2

3

a3(p0

3)=2

4

a4(p0

4)=1.5

Figure 3: The DAG for the second scenario where all prices are ﬁxed to (ln(4),ln(4),ln(3),ln(2))b

and the threshold is t= 0.5. Only product 1 dominates product 4.

The following deﬁnition is an extension of the deﬁnition of a consideration set given an assort-

ment Swhen each product ihas a price pi.

Deﬁnition 5. Given an assortment S, a price vector p= (p1, p2, . . . , pn) and a threshold t, the

consideration set c(S, p) for the Threshold Luce model is deﬁned as:

c(S, p) = {j∈S| 6 ∃i∈S:ai(pi)>(1 + t)aj(pj)}.(9)

The inﬂuence of the price vector over the dominance relations is given by the following example:

Example 5. [Price eﬀect on the dominance relation] Consider the Threshold Luce model deﬁned

over X={1,2,3,4}with utilities u1= ln(10), u2= ln(8), u3= ln(6) and u4= ln(3), and consider

ﬁrst a scenario where all products have the same price pi= ln(3) ∀i= 1,...,4. Consider also

a second scenario with prices equal to p0

1= ln(4), p0

2= ln(4), p0

3= ln(3) and p0

4= ln(2). For a

threshold t= 0.5, we have that ijiﬀ ai(pi)>1.5aj(pj). A table summarizing the utilities, prices,

and attractiveness for both scenarios is given in Table 1 and the DAGs depicting the dominance

relations for the two scenarios are given in Figures 2 and 3.

20

It is also necessary to update the deﬁnition of ρin Deﬁnition 1, since it now depends on the

price of all products in the assortment. The deﬁnition of ρ:X∪ {0} × 2X×(R+∪ ∞)n→[0,1]

becomes:

ρ(i, S, p) =

ai(pi)

Pj∈c(S,p)aj(pj)+a0,if i∈c(S, p),

0 if i /∈c(S, p).

(10)

where a0is the attractiveness of the outside option.

The expected revenue (ER) of an assortment S⊆Xand a price vector p∈Rn

+is given by

R(S, p) = X

i∈c(S,p)

ρ(i, S, p)pi.(ER)

A pair (S, p) with S⊆Xand p∈(R+∪ ∞)nis valid if S={i:pi<∞} and c(S, p) = S. Let

Vbe the set of all valid pairs (S, p). Observe that one can always restrict the search for optimal

solutions to V. Indeed, all dominated products can be given an inﬁnite price and removing them

from the original assortment yields the exact same revenue.

The Joint Assortment and Pricing problem aims at ﬁnding a set S∗and a price vector p∗

satisfying

(S∗, p∗)∈argmax

(S,p)∈V

R(S, p)

and yielding an optimal revenue of

R∗=R(S∗, p∗).

First observe that the strategy used to solve this problem under the multinomial logit does not

carry over to the Threshold Luce Model. Under the multinomial logit, the optimal solution for

the joint assortment and pricing problem is a ﬁxed adjusted margin policy (Wang, 2012) which,

for equal price sensitivities and normalised costs, translates to a ﬁxed price policy. As shown in Li

and Huh (2011), the optimal solution for the pricing problem under the multinomial logit can be

expressed in closed form using the Lambert function W(x) : [0,∞)→[0,∞) which is deﬁned as

the unique function satisfying:

x=W(x)eW(x)∀x∈[0,∞).(11)

Using this function, the optimal revenue can be expressed as:

R∗=WPi∈Xexp(ui−1)

a0(12)

The prices are all equal and satisfy: pi= 1 + R∗∀i∈X. The following example shows that

ﬁxed-price policy is not optimal under the Threshold Luce Model.

Example 6 (Fixed-Price policy is not optimal).Consider 11 products with product 1 having utility

u= 2 and all remaining 10 products having utility u0= 1. Consider a0= 1 and t= 1. Observe

that, for any ﬁxed price, product 1 always dominates the other 10 products having lower utility, as

21

exp(u−u0) = exp(1) = e > (1 + t) = 2. Therefore, the optimal revenue for a a ﬁxed price strategy

is:

Rfixed =Wexp(u−1)

a0=W(e)=1.

As a result, the 10 lower utility products are completely ignored and only product 1 contributes to

the revenue.

Consider the following price scheme now: let the price for product 1 be p= 1.8 and let the

price be p0= 1.4 for the remaining products. Product 1 does not dominate any other product now.

Indeed, for any 1 < k ≤11,

a1

ak

= exp((u−p)−(u0−p0)) = exp((2 −1.8) −(1 −1.4)) ≈1.822 <1 + t= 2,

which yields a revenue of:

R0=p·exp(u−p) + 10 ·p0exp(u0−p0)

exp(u−p) + 10 ·exp(u0−p0) + a0

=1.8·exp(2 −1.8) + 10 ·1.4 exp(1 −1.4)

exp(2 −1.8) + 10 ·exp(1 −1.4) + 1 ≈1.298,

This pricing scheme improves upon the ﬁxed-price policy, yielding a revenue almost %30 higher.

The intuition behind this example is as follows: For a ﬁxed price strategy, the only factor

aﬀecting dominance is the intrinsic utilities because the prices vanish when calculating the ratio

between two attractiveness. This means that the solution can potentially miss the beneﬁts of low

attractiveness products which are dominated by the most attractive product.

It is thus important to understand the structure of an optimal solution for the Joint Assortment

and Pricing problem under the Threshold Luce model. The ﬁrst result states that, for any optimal

solution (S∗, p∗), all product prices are greater or equal than R∗, where R∗denotes the revenue

achieved at optimality.

Proposition 1. In any optimal solution (S∗, p∗), for all i∈S∗,p∗

i≥R∗.

The proof is by contradiction: Removing products with a price lower than R∗yields a greater

revenue. The next proposition characterises the optimal assortment of products of any optimal

solution to the Joint Assortment and Pricing problem. Recall that the products are indexed by

decreasing utility ui. Thus, the set of products [k] := {1, . . . , k}, (with 0 < k ≤n) is said to be an

intrinsic utility ordered set. The following proposition holds:

Proposition 2. Let (S∗, p∗)denote an optimal solution. Then S∗= [k]for some k≤n.

The following Lemma due to Wang and Sahin (2018) is useful to prove some of the upcoming

propositions. For completeness, its proof is also in Appendix A.

Lemma 2 (Lemma 1, Wang and Sahin (2018)).Let H(pi, pj) := pi·exp(ui−pi) + pj·exp(uj−pj),

where exp(ui−pi) + exp(uj−pj) = T. Then, H(pi, pj)is strictly unimodal with respect to pior

pj, and it achieves the maximum at the following point:

p∗

i=p∗

j= ln ((exp(ui) + exp(uj))/T ) (13)

22

Observe that setting the price of a product to ∞is equivalent to not showing it to consumers.

By Proposition 2, one can always ﬁnd an optimal solution that is intrinsic utility ordered. Given a

price vector p∈Rn, let γ(p) : Rn→[n] be deﬁned as γ(p).

=maxi∈[n]is.t pi<∞. Intuitively,

this is the last non-inﬁnite price. Proposition 3 shows that, at optimality, the ﬁnite prices are

non-increasing in i, meaning that lower prices are assigned to lower utility products.

Proposition 3. The prices at an optimal solution (S∗, p∗)satisfy p∗

i≥p∗

i+1 ∀i∈[γ(p)−1].

Moreover, if i, j ∈S∗satisfy ui=uj, then p∗

i=p∗

j.

Recall that the net utility of product iwas deﬁned as: vi=ui−pi. The following proposition

shows that at optimality, net utility follows the same order as intrinsic utility.

Proposition 4. Let p∗be the price of an optimal solution of the Joint Assortment and Pricing

Problem. The following condition holds: ui−p∗

i≥ui+1 −p∗

i+1 ∀i∈[γ(p)−1].

The above propositions make it possible to ﬁlter out non-eﬃcient assortments and prices by

restricting the search space to intrinsic utility ordered assortments and providing insights on how

the optimal solution behaves regarding prices and their relation with utilities. Based on these

propositions, the joint assortment and pricing optimisation problem for the TLM can be written

in a more succinct way. From Proposition 2, the solution is an intrinsic utility ordered set Sk= [k]

for some k≤n. Suppose there exists an optimal solution in the form (Sk, p) for a ﬁxed value k.

In that case, recall that it is suﬃcient to restrict to valid pairs (Sk, p), meaning that c(Sk, p) = Sk.

Consider a ﬁxed k≤n. By Proposition 4, at optimality, ui−pi≥uj−pj∀1≤i < j ≤k.

Therefore, the condition that c(Sk, p) = Skcan be written as

gij (p) := exp(ui−pi)−(1 + t)·exp(uj−pj)≤0,∀1≤i < j ≤k(14)

As a result, the joint k-assortment and pricing optimisation problem for the TLM (JAPTLM-k),

which aims at ﬁnding an optimal assortment Skof size kwith k≤n, can be written as:

maximize

pR(k)(p) := Pi∈Skpi·exp(ui−pi)

Pi∈Skexp(ui−pi) + a0

subject to gij (p)≤0,∀1≤i < j ≤k

(JAPTLM-k)

Note that, if exp(u1−uk)≤(1 + t), then the solution is the same as the unconstrained case,

because any ﬁxed price can be assigned without creating dominances. Hence, the optimal revenue

R(k)can be calculated using equation (12), and all prices are equal to 1 + R(k). On the other

hand, if exp(u1−uk)>1 + t, as in Example 6, the prices need to be adjusted in order to avoid

dominances.

The next theorem is the main result of this section.

Theorem 7. Problem JAPTLM-k can be solved in polynomial time.

The intuition behind the proof is based on Proposition 4 and the study of the Lagrangean

relaxation of problem (JAPTLM-k). Observe that, since ui−pi≥uj−pj(i≤j) at optimality,

23

then the largest ratio between attractiveness is obtained for products 1 and k. This ratio can also

occur for more products but only if they have the same net utility as products 1 or k. Thus, it must

be the case that there are non-negative integers k1and k2with k1+k2≤k, such that letting I1= [k1]

and I2={k−k2+ 1, k −k2+ 2, . . . , k}, the set of constraints C(k1, k2) = {gij(p)|i∈I1, j ∈I2}

are satisﬁed at equality for the optimal solution (see the proof in Appendix A for details). Since it

is only necessary to study a polynomial number of combinations of constraints satisﬁed at equality

and, for each one of those combinations a closed form solution is provided, the result follows.

For the non-trivial case with exp(u1−uk)>1 + t, where a ﬁxed price fails to be optimal, the

prices need to be adjusted in order to avoid the dominances. Let R(k)and p(k)be the optimal

revenue and price vector. The following Lemma characterizes the structure of the optimal solution

for problem JAPTLM-k.

Lemma 3. The optimal solution to problem (JAPTLM-k)is either the same as the unconstrained

case (i.e. ﬁxed price, in the case that exp(u1−uk)≤(1 + t)) or the following holds at optimality:

a1(p1)

ak(pk)= 1 + t. (15)

Moreover, there are non-negative integers k∗

1, k∗

2, with k∗

1+k∗

2≤ksuch that:

R(k)=W

k∗

1+k∗

2

1+t·exp (1+t)Pi∈I1ui+Pi∈I2ui+k∗

2ln(1+t)

k∗

1(1+t)+k∗

2−1+Pi∈¯

Ikexp(ui−1)

a0

,

where I1= [k∗

1],I2={k−k∗

2+ 1, k −k∗

2+ 2, . . . , k}and ¯

Ik= [k]\(I1∪I2). The optimal prices can

be obtained as follows:

p(k)

i=

1 + R(k)+ui−(1+t)Pi∈I1ui+Pi∈I2ui+k∗

2ln(1+t)

k∗

1(1+t)+k∗

2if i∈I1,

1 + R(k)+ui−(1+t)Pi∈I1ui+Pi∈I2ui+k∗

2ln(1+t)

k∗

1(1+t)+k∗

2+ ln(1 + t)if i∈I2,

1 + R(k)if i∈¯

Ik.

(16)

Let TLM-Opt(X, u, a0, k)be the procedure to obtain the optimal solution for problem (JAPTLM-k).

Using TLM-Opt(X, u, a0, k)at most ntimes (once for each k≤n) to obtain the assortment and

prices yielding the highest R(k), one can ﬁnd the optimal assortment and price vector for any given

instance. Its intuition is to mimic the optimal strategy for the regular MNL (Fixed-Price Policy)

as much as possible. However, given that it needs to accommodate prices in order to avoid domi-

nances, the algorithm adjusts prices for the higher intrinsic utility products (making prices larger,

hence less attractive) and reduces the price of lower intrinsic utility ones, making them more attrac-

tive for customers and preventing them from being dominated. This allows the optimal strategy

to have an edge over strategies ignoring the Threshold induced dominances, such as Fixed-Price

Policy and, to a lesser extent, the Quasi-Same Price (Wang and Sahin, 2018). The Quasi-Same

Price policy policy only adjusts the price of the lowest attractiveness product, instead of adjusting

24

both extremes of the attractiveness spectrum and potentially multiple products.

8 Conclusion and Future Work

This paper studies the assortment optimization problem under the Two-Stage Luce model (2SLM),

a discrete choice model introduced by Echenique and Saito (2018) that generalizes the standard

multinomial logit model (MNL) with a dominance relation and may violate regularity. The paper

proved that the assortment problem under the 2SLM can be solved in polynomial time. The paper

also considered the capacitated assortment problem under the 2SLM and proved that the problem

becomes NP-hard in this setting. We also provide polynomial-time algorithms for special cases

of the capacitated problem when (1) the dominance relation is utility-correlated and when (2)

its transitive reduction is a forest. We also provide an Appendix showing numerical experiments

to highlight the performance of the proposed algorithms against classical strategies used in the

literature.

There are at least ﬁve interesting avenues for future research. First, one may wish to study how

to generalize the 2SLM further while still keeping the assortment problem solvable in polynomial

time. For example, one can try to check whether there exists a model that uniﬁes the 2SLM and

the elegant work in Davis, Gallego, and Topaloglu (2013) where the assortment problem is still

solvable in polynomial time. Second, given that the capacitated version of the 2SLM is NP-hard

under Turing reductions (Theorem 4), it is interesting to see whether there exist good approximation

algorithms for this problem. Third, one can explore diﬀerent forms of dominance. For example, one

may consider dominances speciﬁed by a discrete relation or a continuous functional form between

products. Fourth, one can try to generalise our results for the Joint Assortment Pricing Problem

under the Threshold Luce model to a more general setting, where price sensitivities depend on each

product. Finally, one can try to mix attention models with dominance relations, meaning that a

customer ﬁrst perceives a subset of the products, dictated by an attention ﬁlter, and then ﬁlter the

products even more using dominance relations.

9 Acknowledgements

We thanks Yuval Filmus for his helpful insights leading us to ﬁnd useful literature on this topic.

Thanks are also due to Guillermo Gallego for suggesting extending our assortment results to the

GAM model, and to Flavia Bonomo for relevant discussions..

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A Proofs

In this section we provide the proofs missing from the main text.

Proof of Theorem 4. The proof considers four problems:

1. Problem (MWBISBP): Maximum weighted independent set of size at most Cfor bipartite

graphs.

2. Problem (MWEBISBP): Maximum weighted independent set of size equal to Cfor bipartite

graphs.

3. Problem (EC2SLMAP): Optimal assortment under the General Luce model of size C.

4. Problem (C2SLMAP): Optimal capacitated assortment under the Two-Stage Luce model of size

at most C.

The proof shows that Problems (MWEBISBP), (EC2SLMAP), and (C2SLMAP) are NP-hard, using the

NP-hardness of Problem (MWBISBP) (Kalra et al., 2017) as a starting point.

First observe that Problem (MWEBISBP) is NP-hard under Turing reductions. Indeed, Problem

(MWBISBP) can be reduced to solving Cinstances of Problem (MWEBISBP) with budget c(1 ≤c≤C).

We now show that Problem (EC2SLMAP) is NP-hard. Consider Problem (MWEBISBP) over a

bipartite graph G= (V=V1∪V2, E), where V1∩V2=∅, every edge (v1, v2)∈Esatisﬁes

v1∈V1and v2∈V2,wvis the weight of vertex v, and Cis the budget. We show that Problem

(MWEBISBP) over this bipartite graph can be polynomially reduced to Problem (EC2SLMAP). The

reduction assigns each vertex vto a product with a(v) = 1 and rv=wv, sets a0= 0, and has a

capacity C. Moreover, the reduction uses the following dominance relation: v1v2iﬀ (v1, v2)∈E.

This dominance relation is irreﬂexive, anti-symmetric, and transitive, since the graph is bipartite. A

solution to Problem (MWEBISBP) is a feasible solution to Problem (EC2SLMAP), since the independent

set cannot contain two vertices v1, v2with v1v2by construction. Similarly, a feasible assortment

is an independent set, since the assortment cannot select two vertices v1∈V1and v2∈V2with

(v1, v2)∈E, since v1v2. The objective function of Problem (EC2SLMAP) reduces to maximizing

1

CX

v∈V

rvxv

which is equivalent to maximizing Pv∈Vrvxvsince exactly Cproducts will be selected by every

feasible assortment. The result follows by the NP-hardness of Problem (MWEBISBP).

Finally, Problem (C2SLMAP) is NP-hard under Turing reductions. Indeed, Problem (C2SLMAP)

can be reduced to solving Cinstances of Problem (EC2SLMAP) with capacity c(1 ≤c≤C).

Proof of Theorem 5 By Theorem 2, it suﬃces to show that Problem (C-MaxAtt) is solved by

the recurrences in polynomial time. The correctness of recurrence A(v , c) comes from the fact that

vertex vdominates all its descendants and cannot be present in any assortment featuring any of

them. The correctness of recurrence A+(S, c) follows from the fact that eis not dominated by, and

i

does not dominate, any element in S, since they are all children of the same node. This also holds

for the descendants of eand the descendants of the elements in S. Hence, the optimal assortment is

obtained by splitting the capacity cinto n1and n2and merging the best assortment for A+(S, n1)

and A(e, n2) for some n1, n2≥0 summing to c. The recurrences can be solved in polynomial time

since the computation for each vertex vand capacity ctakes O(n C) time, giving an overall time

complexity of O(n2C2).

Proof of Proposition 1. We prove this by contradiction. Suppose p∗

i< R∗for some i∈S, then

ˆ

S=S∗\ {i}has better revenue than the optimal solution if we keep the same prices and p∗

i< R∗.

Indeed, let us calculate R(ˆ

S):

R(ˆ

S) = Pj∈ˆ

Seuj−p∗

j·p∗

j

Pj∈ˆ

Seuj−p∗

j+a0

R(ˆ

S) =Pj∈S∗euj−p∗

j·p∗

j−eui−p∗

i·p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0

R(ˆ

S) = Pj∈S∗euj−p∗

j·p∗

j

Pj∈S∗euj−p∗

j+a0

·Pj∈S∗euj−p∗

j+a0

Pj∈S∗euj−p∗

j−eui−p∗

i+a0

−eui−p∗

i·p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0

R(ˆ

S) = Pj∈S∗euj−p∗

j·p∗

j

Pj∈S∗euj−p∗

j+a0

·"1 + eui−p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0#−eui−p∗

i·p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0

R(ˆ

S) =R∗·"1 + eui−p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0#−eui−p∗

i·p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0

R(ˆ

S) =R∗+eui−p∗

i

Pj∈S∗euj−p∗

j−eui−p∗

i+a0

·[R∗−p∗

i]

| {z }

Γ

Now Γ is positive because p∗

i< R∗, but this implies R(ˆ

S)> R∗, contradicting the optimality

of R∗.

Proof of Proposition 2. Let (S∗, p∗) be an optimal solution. We can assume that (S∗, p∗)∈ V.

We proceed by contradiction. Suppose that there is a product inot included in the optimal solution

and another product jwith smaller intrinsic utility included in S∗. We show that we can include

product i, and remove jand get a greater revenue. Let ˆ

S= (S∗\ {j})∪ {i}, be the set where

we removed product j, and included product i. Let ˆpi=ui−uj+p∗

j, this means that the total

attractiveness remains unchanged, and no new domination relations appear, given that product j

already had the same level attractiveness that product inow has. Observe that given that ui≥uj,

we have that ˆpi≥p∗

j. Let us calculate R(ˆ

S, ˆp), where ˆpis the same as p∗, but with the proposed

changes in price:

ii

R(ˆ

S, ˆp) = Pk∈ˆ

Seuk−ˆpk·ˆpk

Pk∈ˆ

Seuk−ˆpk+a0

R(ˆ

S, ˆp) = Pk∈S∗euk−p∗

k·p∗

k−euj−p∗

j·p∗

j+eui−ˆpi·ˆpi

Pk∈ˆ

Seuk−ˆpk+a0

R(ˆ

S, ˆp) = Pk∈S∗euk−p∗

k·p∗

k

Pk∈S∗euk−ˆpk+a0

| {z }

R∗

+eui−ˆpi·ˆpi−euj−p∗

j·p∗

j

Pk∈ˆ

Seuk−ˆpk+a0

R(ˆ

S, ˆp) = R∗+euj−p∗

j

Pk∈ˆ

Seuk−ˆpk+a0

| {z }

≥0

·ˆpi−p∗

j

| {z }

>0

R(ˆ

S, ˆp)> R∗

Where we ﬁrst rewrite R(ˆ

S, ˆp) using (S∗, p∗) because we just swapped product ifor product

j, and the total attractiveness remain the same, so the denominator does not change. Then we

identify R(S, p), and we use ui−ˆpi=uj−pjto being able to factorize the remaining terms. So

we found a pair ( ˆ

S, ˆp), yielding strictly more revenue than (S, p), but adding product i, which

contradicts the optimality of (S∗, p∗).

Proof of Lemma 2. The proof (due to Wang and Sahin (2018)) is useful because it provides

intuition on how the optimal price variates when constrained to a ﬁxed additive market share

among any two products. By the equality constraint, we have pj=uj−ln(T−exp(ui−pi)), so

H(pi, pj) can be rewritten purely as a function of pias:

H(pi) = pi·exp(ui−pi)+(uj−ln(T−exp(ui−pi))) ·(T−exp(ui−pi)).(17)

Now, let us calculate the ﬁrst derivative of H(pi) w.r.t. pi:

∂H (pi)

∂pi

= (−pi+ (uj−ln(T−exp(ui−pi)))) ·exp(ui−pi) (18)

Clearly the left-hand side term on the multiplication is monotonically decreasing from positive to

negative values as piincreases from 0 to ∞. Therefore H(pi) is strictly unimodal and reaches its

maximum value at:

p∗

i=p∗

j= ln ((exp(ui) + exp(uj))/T ).

Proof of Proposition 3. We prove this result by contradiction. Let ibe the ﬁrst index where

this condition does not hold, this means that p∗

i< p∗

i+1. Using Lemma 2, we found ˆpsatisfying

p∗

i<ˆp < p∗

i+1. Does this new price alter the consideration set? We show that this is not the case.

Indeed, the eﬀect is two-fold: the price for product iincreases, and the price for product i+ 1

iii

decreases. We analyse the eﬀect of these two consequences:

•Increase on price for product i: This means a(i, p) decreases. Note that ui−ˆp≥ui+1 −p∗

i+1,

so neither ii+ 1 or i+ 1 i, because their attractiveness are now even closer than before.

Can ibe dominated now by another product? No, because given that ui≥ui+1 we have

ui−ˆp≥ui+1 −ˆp≥ui+1 −p∗

i+1. Therefore the new attractiveness of iis still larger than the

new attractiveness of i+ 1, and the last inequality implies that the new attractiveness of iis

larger than the old attractiveness of i+ 1, and i+ 1 was not previously dominated either by

any other product.

•Decrease on price for product i+ 1: Previously i+ 1 was not dominated by any product. Can

i+ 1 be dominated now? No, because if i+ 1 was not dominated before, now with a smaller

price ˆpits attractiveness is larger and therefore can’t be dominated now either (the only other

product that changed attractiveness was i, and it now has smaller attractiveness). Can i+ 1

dominate another product now with its new higher attractiveness? No, because given that

ui≥ui+1 we have ui−p∗

i≥ui+1 −p∗

i≥ui+1 −ˆp, so the old attractiveness of product i

is larger than the new attractiveness of product i+ 1, and given that idid not dominate

another product before, the new price does not make i+ 1 dominate another product either.

So, letting pfix exactly the same as p∗, but replacing both p∗

iand p∗

i+1 with ˆp, means that the

pair (S∗, pfix) yields strictly more revenue than (S∗, p∗) (by Lemma 2), contradicting the optimality

assumption. The fact that equal intrinsic utility implies equal price at optimality, can be easily

demonstrated by the following: if two equal intrinsic utility products have diﬀerent prices, then

using Lemma 2 we obtain strictly better revenue by assigning them the same price, and no new

domination occurs, because the new price is conﬁned between the previous prices.

Proof of Proposition 4. We prove this by contradiction. Let p∗be the optimal solution and i

be the ﬁrst index where this condition does not hold. This means that ui−p∗

i< ui+1 −p∗

i+1. We

can extrapolate this inequality further and say:

ui+1 −p∗

i< ui−p∗

i< ui+1 −p∗

i+1 < ui−p∗

i+1,(19)

because ui≥ui+1 and pi≥pi+1 by Propositions 2 and 3 respectively. We now do the following:

Deﬁne p0

iand p0

i+1 such as exp(ui−p0

i) + exp(ui+1 −p0

i+1) = exp(ui−p∗

i) + exp(ui+1 −p∗

i+1) and

exp(ui−p0

i) = exp(ui+1 −p0

i+1). This means that:

p0

i=ui−ln exp(ui−p∗

i) + exp(ui+1 −p∗

i+1)

2

p0

i+1 =ui+1 −ln exp(ui−p∗

i) + exp(ui+1 −p∗

i+1)

2

Consider H(pi, pi+1) = pi·exp(ui−pi) + pj·exp(ui+1 −pi+1 ), where exp(ui−pi) + exp(ui−

pi) = exp(ui−p∗

i) + exp(ui+1 −p∗

i+1). By Lemma 2, H(pi, pi+1) is strictly increasing in pifor

iv

pi≤ˆpand strictly decreasing for pi≥ˆp, with ˆp= ln exp(ui)+exp(ui+1)

exp(ui−p∗

i)+exp(ui+1−p∗

i+1)the solution of

the corresponding maximization problem of Lemma 2. We can verify that ˆp < p0

i< p∗

i. The ﬁrst

inequality is straightforward. Indeed:

p0

i=ui−ln exp(ui−pi) + exp(ui+1 −pi+1)

2

p0

i= ln 2 exp(ui)

exp(ui−p∗

i) + exp(ui+1 −p∗

i+1)

p0

i>ln exp(ui) + exp(ui+1)

exp(ui−p∗

i) + exp(ui+1 −p∗

i+1)

| {z }

ˆp

p0

i>ˆp

proving the desired inequality. Now, for the second one:

p0

i=ui−ln exp(ui−pi) + exp(ui+1 −pi+1)

2

p0

i= ln 2 exp(ui)

exp(ui−p∗

i) + exp(ui+1 −p∗

i+1)

p0

i≤ln 2 exp(ui)

exp(ui−p∗

i) + exp(ui−p∗

i+1)

p0

i= ln 2 exp(ui)

exp(ui)(exp(−p∗

i) + exp(−p∗

i+1))

p0

i<ln 2

2 exp(−p∗

i)

p0

i< p∗

i,

thus we have:

p0

i·exp(ui−p0

i) + p0

i+1 ·exp(ui+1 −p0

i+1)> p∗

i·exp(ui−p∗

i) + p∗

i+1 ·exp(ui+1 −p∗

i+1).

Meaning that we have the same assortment, but with prices p0

iand p0

i+1 generating strictly more

revenue than the optimal prices, which is a contradiction. The only thing that we have left to show

that with these new prices we are still on the same consideration set. It would be enough to show

that the new net utilities are bounded by previous values of net utilities. Indeed, we can verify

that p∗

i+1 ≤p0

i+1 ≤p0

i≤p∗

i, by simply using the deﬁnitions. We also know, by hypothesis that

ui−p0

i=ui+1 −p0

i+1, then ui−p0

i=ui+1 −p0

i+1 ≤ui+1 −p∗

i+1. So even when the price of product

idecreased, the new attractiveness is bounded above by a previously existing attractiveness, thus

not changing the consideration set. By the same reasoning, ui+1 −p0

i+1 =ui−p0

i≥ui−p∗

i, meaning

v

that the new attractiveness is bounded below by a pre-existing one, so i+ 1 is not dominated with

this new prices either. So the consideration set stays the same, concluding the proof.

Proof of Theorem 7. We ﬁrst write problem (JAPTLM-k) in minimization form to directly apply

the Karush-Khun-Tucker conditions (KKT)(Karush, 1939).

minimize

p−R(k)(p)

subject to gij (p)≤0,∀1≤i<j≤k

(20)

The associated Lagrangean function is:

Lk(p, µ) = −R(k)(p) + X

1≤i<j≤k

µij ·gij (p),(21)

where µij ≥0 are the associated Lagrange multipliers. Recall that if exp(u1−uk)≤(1 + t), the

optimal revenue R(k)can be calculated using equation (12), and the solution corresponds to a ﬁxed

price policy as for the regular multinomial logit.

On the other hand, if exp(u1−uk)>(1 + t), any ﬁxed price causes product kto be dominated

by product 1. Thus, to include product kin the assortment we need to adjust the prices. Let

p= (p1, . . . , pk) be the optimal price vector for problem (20). Observe that it can’t happen that

a1(p1)

ak(pk)<1+t, since by Proposition 4, it will also means that a1(p1)

a2(p2)<1+tand using Lemma 2 we can

ﬁnd ˆpsuch that assigning ˆpto products 1 and 2 yields a larger revenue (and no dominance relation

appears, since the attractiveness of product 1 was reduced, and the attractiveness of product 2

increased, but is still less than the one of product 1), which contradicts optimality. Therefore, g1k

must be satisﬁed with equality, meaning a1(p1)

ak(pk)= 1 + t.

Furthermore, at optimality it holds ui−pi≥uj−pj∀i≤j(by Proposition 4), and thus the

biggest ratio between attractiveness is observed for products 1 and k, and is exactly equal to 1 + t.

This ratio can be replicated for other pairs of products, but only if they share the same net utility

(and thus attractiveness) to the one of products 1 or k. Therefore, it must be the case that there are

integers k1and k2with k1+k2≤k, such that all products in I1= [k1] share the same attractiveness

(a1(p1)) and all products in I2={k−k2+ 1, k −k2+ 2, . . . , k}share the same attractiveness as

well (ak(pk)). This means that the set of constraints C(k1, k2) = {gij(p)|i∈I1, j ∈I2}are all

satisﬁed with equality at optimality.

We now study the derivative of equation (21) with respect to each price pito obtain the KKT

conditions. We here assume that the ﬁrst k1values share the same net utility value, meaning

us=u1−p1=ui−pi∀i∈I1, and for the last k2products, we also have the same value of net

utility, that we call uf, this is: uf=uk−pk=ui−pi∀i∈I2. Where these two quantities satisfy:

us−uf= ln(1 + t),

Let us write the derivatives of the Lagrangean depending on where the index ibelongs. If i∈I1,

vi

then:

dLk

dpi

=exp(ui−pi)

Pj∈Skexp(uj−pj) + a0

·hpi−1−R(k)(p)i−exp(ui−pi)·X

j∈I2

µij ,(22)

if i∈I2, we have:

dLk

dpi

=exp(ui−pi)

Pj∈Skexp(uj−pj) + a0

·hpi−1−R(k)(p)i+ (1 + t) exp(ui−pi)·X

j∈I1

µji ,(23)

And ﬁnally, if i∈¯

Ik= [k]\(i1∪I2), the derivative takes the following form:

dLk

dpi

=exp(ui−pi)

Pj∈Skexp(uj−pj) + a0

·hpi−1−R(k)(p)i(24)

Observe that ∀i∈¯

Ik,dLk

dpi= 0 =⇒pi= 1 + R(k)(p), and the right hand side is not dependent

on i, so all products in ¯

Ikshare the same price, which we denote ¯p. We can rewrite all prices and

the revenue depending on usand ¯p, using the following relations:

1. ∀i∈I1u1−p1=ui−pi=⇒pi=ui−us

2. ∀i∈I2u1−p1=ui−pi+ ln(1 + t) =⇒pi=ui−us+ ln(1 + t)

Note now that at optimality, for a ﬁxed k, prices are determined by k1and k2. Thus, the

optimal revenue can be written explicitly depending on k,k1and k2, taking the following form:

R(k)(k1, k2) =

Pi∈I1(ui−us) exp(us) + ¯pexp(−¯p)Pi∈¯

Ikexp(ui) + Pi∈I2(ui−us+ ln(1 + t)) exp(us−ln(1 + t))

Pi∈I1exp(us) + exp(−¯p)Pi∈¯

Ikexp(ui) + Pi∈I2exp(us+ ln(1 + t)) + a0

(25)

Note that ¯p= 1 + R(k)(k1, k2) (Equation (24)) and let E(k1, k2) = Pi∈¯

Ikexp(ui). Using these

two relations, we can rewrite the optimal revenue as:

R(k)(k1, k2) =

eusP

i∈I1

(ui−us) + eus

1+t·P

i∈I2

(ui−us+ ln(1 + t)) + E(k1, k2)(1 + R(k)(k1, k2))e−(1+R(k)(k1,k2))

eushk1+k2

1+ti+E(k1, k2)e−(1+R(k)(k1,k2)) +a0

(26)

Up to this point, we have an equation relating the optimal revenue R(k)(k1, k2) and us. From

equation (22), after reordering terms we have:

pi−1−R(k)(k1, k2)

eus(k1+k2(1 + t)) + E(k1, k2)e−(1+R(k)(k1,k2)) +a0

=X

j∈I2

µij ,∀i∈I1

ui−us−1−R(k)(k1, k2)

eus(k1+k2(1 + t)) + E(k1, k2)e−(1+R(k)(k1,k2)) +a0

=X

j∈I2

µij ,∀i∈I1(27)

vii

Analogously, from equation (23), after reordering terms we have ∀i∈I2:

pi−1−R(k)(k1, k2)

eus(k1+k2(1 + t)) + E(k1, k2)e−(1+R(k)(k1,k2)) +a0

=−(1 + t)X

j∈I1

µji ,∀i∈I2

1

1 + t·ui−us+ ln(1 + t)−1−R(k)(k1, k2)

eus(k1+k2(1 + t)) + E(k1, k2)e−(1+R(k)(k1,k2)) +a0

=−X

j∈I1

µji ,∀i∈I2(28)

Now, if we add equations (27) ∀i∈I1then take equations (28) and also add them ∀i∈I2, and

add those two results we can derive the value R(k)(k1, k2) as follows.

X

i∈I1

j∈I2

µ